Finite Group Databasebeta

← #37571648

Subgroups — #37571648

Order 24448.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133333C211yesmaximal, characteristic
2#27133035C211yesmaximal, characteristic, complemented
3#27132959C211yesmaximal, characteristic, complemented
4#27132954C211yesmaximal, characteristic, complemented
5#27132393C211yesmaximal, characteristic, complemented
6#27132390C211yesmaximal, characteristic, complemented
7#331×C191C211yesmaximal, characteristic, complemented
8C2×#15690286—21no—
9C2×#15690275—22no—
10#20259322C2211yescharacteristic
11#20259318—21no—
12#20259294—22no—
13#20259293—21no—
14C4×Dic382C2211yescharacteristic
15C4×D764C2211yescharacteristic
16D4×Dic191—21no—
17#20259256—21no—
18#20259254—21no—
19#20259247—21no—
20#20259244—21no—
21C4×C191⋊C8C2211yescharacteristic
22#127×C191C2211yescharacteristic
23C4⋊C8×C191C2211yescharacteristic
24C4⋊C8×C191C2211yescharacteristic
25#15690286—42no—
26#15690286—42no—
27C2×Dic382—21no—
28C2×Dic382D411yescharacteristic
29C2×D764D411yescharacteristic
30C2×C22⋊D191—22no—
31C2×C22⋊D191—22no—
32C22×Dic191—22no—
33C2×C4×D191—21no—
34#15690275—44no—
35C2×C191⋊C8—21no—
36C4⋊Dic191—21no—
37C4⋊Dic191—21no—
38C4⋊Dic191D411yescharacteristic
39C4⋊Dic191D411yescharacteristic
40Dic191⋊C4—21no—
41C22⋊Dic191—22no—
42(C2×D191)⋊C4—21no—
43C4×Dic191—21no—
44C2×D4×C191—21no—
45C2×D4×C191—21no—
46C2×D4×C191D412yes—
47C2×C8×C191—21no—
48C2×C8×C191—21no—
49C42×C191C2311yescharacteristic
50C22×D191—21no—
51Dic382—21no—
52Dic382—42no—
53Dic382SD1612yescomplemented
54D764—21no—
55D764—21no—
56C22⋊D191—44no—
57C22⋊D191—82no—
58C2×Dic191—21no—
59C2×Dic191—21no—
60C2×Dic191—21no—
61C2×Dic191—44no—
62C2×Dic191—41no—
63C4×D191—41no—
64C4×D191—41no—
65C191⋊C8—22no—
66C23×C191—22no—
67D4×C191—42no—
68D4×C191—41no—
69D4×C191—41no—
70D4×C191—41no—
71D4×C191—41no—
72C2×C4×C191C2×D411yescharacteristic
73C2×C4×C191C2×D411yescharacteristic
74C2×C4×C191C2×D411yescharacteristic
75C8×C191—41no—
76C8×C191—41no—
77C22×C191—42no—
78C22×C191—42no—
79C22×C191—42no—
80C22×C191#12011yescharacteristic
81C2×D191—41no—
82C2×D191—41no—
83C2×D191—41no—
84C4×C191—21no—
85C4×C191C2×SD1611yescharacteristic
86C4×C191#13511yescharacteristic
87C4×C191AGL(1,ℤ/8)11yescharacteristic
88C4×C191#13711yescharacteristic
89Dic191—22no—
90Dic191—41no—
91Dic191—41no—
92Dic191—81no—
93C2×C191—82no—
94C2×C191#45011yescharacteristic
95C2×C191#45211yescharacteristic
96C2×C191#45311yescharacteristic
97D191—81no—
98C191#163111yescharacteristic, complemented, Sylow
99#1631—1911nomaximal, Sylow
100#550—1911no—
101#492—1911no—
102#477—1911no—
103#476—1911no—
104#331D19111yescharacteristic, complemented
105Q8⋊C8—1911no—
106D4⋊C8—1911no—
107C2×#41—3821no—
108C2×SD16—3822no—
109C4⋊Q8—1911no—
110#127—3821no—
111#127C2×D19111yescharacteristic
112#121—3822no—
113#121—3821no—
114C4×Q8—1911no—
115C4×D4—3821no—
116C4×D4—1911no—
117#107—3821no—
118#106—3821no—
119C4⋊C8C2×D19111yescharacteristic
120C4⋊C8C2×D19111yescharacteristic
121#103—3821no—
122#102—3821no—
123C4×C8—1911no—
124#41—7642no—
125#41—7642no—
126C2×Q8—3821no—
127C2×Q8—1911no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—3822no—
131C2×D4—3822no—
132C2×D4—1911no—
133C2×D4D76412yes—
134C22×C4—3822no—
135C22×C4—3821no—
136SD16—7644no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—3821no—
140C4⋊C4—3821no—
141C4⋊C4—3821no—
142C4⋊C4—3821no—
143C4⋊C4—1911no—
144C4⋊C4—1911no—
145C22⋊C4—3822no—
146C22⋊C4—3821no—
147C42—3821no—
148C42C22×D19111yescharacteristic
149C23—22no—
150C23—3821no—
151Q8—3821no—
152Q8—1912no—
153Q8—7642no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—3821no—
160D4—3821no—
161D4—15282no—
162D4—7644no—
163C2×C4—3821no—
164C2×C4—3821no—
165C2×C4—3821no—
166C2×C4—7644no—
167C2×C4—7641no—
168C2×C4—7641no—
169C2×C4—7641no—
170C2×C4D4×D19111yescharacteristic
171C2×C4D4×D19111yescharacteristic
172C2×C4C2×D76411yescharacteristic
173C8—41no—
174C8—41no—
175C8—3822no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—7641no—
180C22—7641no—
181C22—7641no—
182C22#2025929211yescharacteristic
183C4—21no—
184C4—3822no—
185C4—15281no—
186C4—7641no—
187C4—7641no—
188C4SD16×D19111yescharacteristic
189C4#2025934111yescharacteristic
190C4#2025934611yescharacteristic
191C4#2025935111yescharacteristic
192C2—82no—
193C2—15281no—
194C2#2713281511yescharacteristic
195C2#2713282611yescharacteristic
196C2#2713283311yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC42×C191C23
FrattiniC42C22×D191
Fitting#331×C191C2
socleC22×C191#120
prime radical#27133333C2
hypercenter#331D191
hypocenterC191#1631
LCSG ▷ C42×C191 ▷ C22×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #27133035C2semidirect
3. #27132959C2semidirect
4. #27132954C2semidirect
5. #27132393C2semidirect
6. #27132390C2semidirect
7. #331×C191C2semidirect
53. Dic382SD16semidirect
98. C191#1631semidirect
104. #331D191semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#271333332
#271330352
#271329592
#271329542
#271323932
#271323902
#331×C1912
#1631191